Exponential distribution is a probability distribution that models the time between
events occurring at a constant rate. It is a continuous distribution with a single
parameter, which is the rate parameter, often denoted as λ.
The probability density function (PDF) of the exponential distribution is given by:
f(x) = λe^(-λx)
where x is the time, and e is the mathematical constant approximately equal to 2.71828.
An example of the exponential distribution is the time between two consecutive arrivals
of customers at a store. Suppose customers arrive at a store at a rate of λ=2 per minute.
Then, the probability that the time between two consecutive arrivals is less than or equal
to t is given by:
P(X ≤ t) = 1 - e^(-2t)
where X is the time between consecutive arrivals.
Another example is the lifetime of a certain type of light bulb. Suppose the lifetime of a
certain type of light bulb follows an exponential distribution with a rate parameter of
λ=0.01 per hour. Then, the probability that a bulb will last for at least 50 hours is:
P(X ≥ 50) = e^(-0.01 * 50) = 0.6065
where X is the lifetime of the bulb in hours.